3.614 \(\int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{(x+1)^{12}}{14 x^{14}}+\frac{(x+1)^{12}}{91 x^{13}}-\frac{(x+1)^{12}}{1092 x^{12}} \]

[Out]

-(1 + x)^12/(14*x^14) + (1 + x)^12/(91*x^13) - (1 + x)^12/(1092*x^12)

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Rubi [A]  time = 0.0236464, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{(x+1)^{12}}{14 x^{14}}+\frac{(x+1)^{12}}{91 x^{13}}-\frac{(x+1)^{12}}{1092 x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((1 + x)*(1 + 2*x + x^2)^5)/x^15,x]

[Out]

-(1 + x)^12/(14*x^14) + (1 + x)^12/(91*x^13) - (1 + x)^12/(1092*x^12)

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Rubi in Sympy [A]  time = 6.54921, size = 29, normalized size = 0.78 \[ - \frac{\left (x + 1\right )^{12}}{1092 x^{12}} + \frac{\left (x + 1\right )^{12}}{91 x^{13}} - \frac{\left (x + 1\right )^{12}}{14 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)*(x**2+2*x+1)**5/x**15,x)

[Out]

-(x + 1)**12/(1092*x**12) + (x + 1)**12/(91*x**13) - (x + 1)**12/(14*x**14)

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Mathematica [B]  time = 0.00403979, size = 79, normalized size = 2.14 \[ -\frac{1}{14 x^{14}}-\frac{11}{13 x^{13}}-\frac{55}{12 x^{12}}-\frac{15}{x^{11}}-\frac{33}{x^{10}}-\frac{154}{3 x^9}-\frac{231}{4 x^8}-\frac{330}{7 x^7}-\frac{55}{2 x^6}-\frac{11}{x^5}-\frac{11}{4 x^4}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^15,x]

[Out]

-1/(14*x^14) - 11/(13*x^13) - 55/(12*x^12) - 15/x^11 - 33/x^10 - 154/(3*x^9) - 2
31/(4*x^8) - 330/(7*x^7) - 55/(2*x^6) - 11/x^5 - 11/(4*x^4) - 1/(3*x^3)

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Maple [A]  time = 0.009, size = 62, normalized size = 1.7 \[ -{\frac{55}{12\,{x}^{12}}}-{\frac{11}{13\,{x}^{13}}}-{\frac{55}{2\,{x}^{6}}}-{\frac{11}{4\,{x}^{4}}}-33\,{x}^{-10}-{\frac{231}{4\,{x}^{8}}}-15\,{x}^{-11}-{\frac{1}{14\,{x}^{14}}}-{\frac{154}{3\,{x}^{9}}}-{\frac{1}{3\,{x}^{3}}}-11\,{x}^{-5}-{\frac{330}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(x^2+2*x+1)^5/x^15,x)

[Out]

-55/12/x^12-11/13/x^13-55/2/x^6-11/4/x^4-33/x^10-231/4/x^8-15/x^11-1/14/x^14-154
/3/x^9-1/3/x^3-11/x^5-330/7/x^7

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Maxima [A]  time = 0.691704, size = 81, normalized size = 2.19 \[ -\frac{364 \, x^{11} + 3003 \, x^{10} + 12012 \, x^{9} + 30030 \, x^{8} + 51480 \, x^{7} + 63063 \, x^{6} + 56056 \, x^{5} + 36036 \, x^{4} + 16380 \, x^{3} + 5005 \, x^{2} + 924 \, x + 78}{1092 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^15,x, algorithm="maxima")

[Out]

-1/1092*(364*x^11 + 3003*x^10 + 12012*x^9 + 30030*x^8 + 51480*x^7 + 63063*x^6 +
56056*x^5 + 36036*x^4 + 16380*x^3 + 5005*x^2 + 924*x + 78)/x^14

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Fricas [A]  time = 0.263914, size = 81, normalized size = 2.19 \[ -\frac{364 \, x^{11} + 3003 \, x^{10} + 12012 \, x^{9} + 30030 \, x^{8} + 51480 \, x^{7} + 63063 \, x^{6} + 56056 \, x^{5} + 36036 \, x^{4} + 16380 \, x^{3} + 5005 \, x^{2} + 924 \, x + 78}{1092 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^15,x, algorithm="fricas")

[Out]

-1/1092*(364*x^11 + 3003*x^10 + 12012*x^9 + 30030*x^8 + 51480*x^7 + 63063*x^6 +
56056*x^5 + 36036*x^4 + 16380*x^3 + 5005*x^2 + 924*x + 78)/x^14

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Sympy [A]  time = 0.583568, size = 61, normalized size = 1.65 \[ - \frac{364 x^{11} + 3003 x^{10} + 12012 x^{9} + 30030 x^{8} + 51480 x^{7} + 63063 x^{6} + 56056 x^{5} + 36036 x^{4} + 16380 x^{3} + 5005 x^{2} + 924 x + 78}{1092 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)*(x**2+2*x+1)**5/x**15,x)

[Out]

-(364*x**11 + 3003*x**10 + 12012*x**9 + 30030*x**8 + 51480*x**7 + 63063*x**6 + 5
6056*x**5 + 36036*x**4 + 16380*x**3 + 5005*x**2 + 924*x + 78)/(1092*x**14)

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GIAC/XCAS [A]  time = 0.268897, size = 81, normalized size = 2.19 \[ -\frac{364 \, x^{11} + 3003 \, x^{10} + 12012 \, x^{9} + 30030 \, x^{8} + 51480 \, x^{7} + 63063 \, x^{6} + 56056 \, x^{5} + 36036 \, x^{4} + 16380 \, x^{3} + 5005 \, x^{2} + 924 \, x + 78}{1092 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^15,x, algorithm="giac")

[Out]

-1/1092*(364*x^11 + 3003*x^10 + 12012*x^9 + 30030*x^8 + 51480*x^7 + 63063*x^6 +
56056*x^5 + 36036*x^4 + 16380*x^3 + 5005*x^2 + 924*x + 78)/x^14